ANÁLISE ECONÓMICA • 52

XOSÉ LUIS QUIÑOÁ LÓPEZ

Universidade de Santiago de Compostela

ABOUT THE SIZE OF THE MEASUREMENT UNITS IN A LEONTIEF SYSTEM AND ITS CONSEQUENCES

CONSELLO EDITOR: José Antonio Aldrey Vázquez Dpto. Xeografía.

Manuel Antelo Suárez Dpto. Fundamentos da Análise Económica.

Juan J. Ares Fernández Dpto. Fundamentos da Análise Económica.

Xesús Leopoldo Balboa López Dpto. Historia Contemporánea e América.

Roberto Bande Ramudo Dpto. Fundamentos da Análise Económica.

Joam Carmona Badía Dpto. Historia e Institucións Económicas.

Luis Castañón Llamas Dpto. Economía Aplicada.

Melchor Fernández Fernández Dpto. Fundamentos da Análise Económica.

Manuel Fernández Grela Dpto. Fundamentos da Análise Económica.

Xoaquín Fernández Leiceaga Dpto. Economía Aplicada.

Lourenzo Fernández Prieto Dpto. Historia Contemporánea e América.

Carlos Ferrás Sexto Dpto. Xeografía.

Mª do Carmo García Negro Dpto. Economía Aplicada.

Xesús Giráldez Rivero Dpto. Historia Económica.

Wenceslao González Manteiga Dpto. Estatística e Investigación Operativa.

Manuel Jordán Rodríguez Dpto. Economía Aplicada.

Rubén C. Lois González Dpto. Xeografía e Historia.

López García, Xosé Dpto. Ciencias da Comunicación

Edelmiro López Iglesias Dpto. Economía Aplicada.

Maria L. Loureiro García Dpto. Fundamentos da Análise Económica.

Manuel Fco. Marey Pérez Dpto. Enxeñería Agroforestal.

Alberto Meixide Vecino Dpto. Fundamentos da Análise Económica.

Miguel Pazos Otón Dpto. Xeografía.

Carlos Ricoy Riego Dpto. Fundamentos da Análise Económica.

Maria Dolores Riveiro García Dpto. Fundamentos da Análise Económica.

Javier Rojo Sánchez Dpto. Economía Aplicada.

Xosé Santos Solla Dpto. Xeografía.

Francisco Sineiro García Dpto. Economía Aplicada.

Ana María Suárez Piñeiro Dpto. Historia I.

ENTIDADES COLABORADORAS - Consello Económico e Social de Galicia - Fundación Feiraco - Fundación Novacaixagalicia-Claudio San

Martín

Edita: Servicio de Publicacións da Universidade de Santiago de Compostela ISSN: 1138-0713 D.L.G.: C-1842-2007

1

About the size of the measurement units in a Leontief system and its consequences

Xosé Luis Quiñoá López1

ABSTRACT

Beginning from the economic meaning of indecomposability in an Leontief-type system

and then modifying the size of the measure units of the different goods through

consecutive approximations, the original system is turned into another, structurally

equivalent, one whose technological matrix A is such that ∀j, aai

ij = , the

maximum eigenvalue of A that admits )1,...,1(1 = as left positive eigenvector, from

where we deduce the main Perron-Frobenius theorem.

Keywords: Leontief system, Units size, Consecutive approximations, Perron-Frobenius.

1 University of Santiago de Compostela, Faculty of Economic and Business Sciences, Department of Quantitative Economy, Northern University Campus – 15782 Santiago de Compostela Phone Number.: 34 881 8 11516 – joseluis.quinoa@usc.es

2

INTRODUCTION

A Leontief system is represented though a board

( ) ==

n

i

n

i

nnnjn

inij

nj

ij

Q

Q

Q

Qqqqqqqqq

q

11

1

1111

; ; = (1)

in which:

– (qij) represents the interindustrial transactions matrix; each term qij shows the

physical quantity of the goods i used by the industry j in the considered period of

time.

– β = (βi) is the column vector that represents the surplus of the system; βi is the

surplus of the industry i, and we suppose that it exists at least one industry in which

βi > 0.

– Q = (Qi) is the column vector that represents the total output of the different goods in

the considered period.

– L = (L1,…, Li,…, Ln) is the row vector of the labour quantities used by the different

industries.

– A = (aij) shows the square matrix n×n defined by j

ijij Q

qa = , and that we will name the

system´s technological matrix.

– l = (li ) is the row vector defined by i

ii Q

Ll = ; li represents the amount of labour used

in the production of a product´s unit i.

– We will denote as lA what we know as the “system´s technique”.

Each column matrix (qij) or (aij) represents heterogeneous goods, whose respective

quantities depend on the measurement units (kilograms, tons, dozens...). From that,

we deduce that the system type (1) can be represented by a countless matrix (qij) or

(aij), we only need to modify the size of one of the measurement units for obtaining a

different matrix.

3

Our target is to find a technological matrix A that could be considered as

“characteristic” in a given Leontief system.

We will see that this is like taking the size of one the measurement units as numeraire,

and according to it, the other units can be expressed. This will be possible for

indecomposable systems, for certain decomposable systems, and even for some infinite

ones.

Let´s consider the example suggested by Sraffa at the beginning of Production of

Commodities by Means of Commodities, relating to two iron and wheat industries,

whose quantities had been measured respectively in tons and quarters. In Leontief

terminology, we represent the problem as

( ) ==20

575 ;

0175

; 812

120280= Qijq (2)

with a technological matrix

≈4,0020869,0

6486956,0A

The technological matrix A is

==i

iaS 507825,011 ; ==i

iaS 4,622

What suggests that the quarter is a measurement unit too small relative to the ton,

making the total numerical quantity of wheat, Q1 = 575, excessively large in

comparison with the iron Q2 = 20.

Let´s suppose that we take 5 quarters for wheat and 31 tons for iron as measurement

units. The system results:

( ) ==60

115 ;

035

' ; 24362456

' = 'Qijq (3)

4

whose technological matrix is:

( )ijaA '4,0313043,04,0486956,0

' =≅

which has the important property that

==i

iaS 8,0'' 11 ; ==i

iaS 8,0'' 22

The system (3) is structurally equivalent to the original (2), with the only difference that

the same physical quantity of the goods is being expressed in different measurement

units, giving as a as result different numerical quantities.

We can see that:

a) 8,0=a is the maximum eigenvalue of A (and A′ ).

b) 3 ,51 (or any possible multiple) of it is the positive eigenvector on the left

associated to the maximum eigenvalue of A, 8,0=a . Likewise, )1,...,1(1 = is the

positive eigenvector on the left of A´ associated to 8,0=a (the wheat units are five

times bigger and the iron ones three times smaller).

c) The Leontief inverse of A′ is:

)(8095,27142,11904,22857,3

)'( 1ijAI =≅− −

and it has the property that

aii −

=≅1

151 ; ai

i −=≅

1152

5

d) In the system (3) the ratio between means of production used ji

ijq,

' and total

production iQ' results

aQ

q

i

jiij

=== 8,0175140

'

',

and we also deduce the ratios according to a between surplus and total output, and

surplus and used production means.

e) Taking == 3 ,51) ,( 21 rrr as eigenvector on the left of A associated to a =0,8, and

=3005/1

r̂ we have 1ˆ ̂' −= rArA .

In the example, as we saw, we take as unit 5 quarters for wheat and 31 tons for iron,

which is the same as saying that in the system 1 ton of iron is equal to 15 quarters of

wheat, and that is the accurate conclusion reached by Sraffa: “The exchange ratio which

the advances to be replaced and the profits to be distributed to both industries in

proportion to their advances is 15 qr. of wheat for 1 t. of iron, and the corresponding

rate of profits in each industry is 25%”.

Regrettably, Sraffa –who seems to have sensed the importance of considering the size

of the measurement units- did not continue on that way, maybe because his efforts were

straightly directed to the distribution problem.

If matrix A is indecomposable and the system has any kind of surplus, then the theorem

of Perron-Frobenius indicates that its maximum eigenvalue is a < 1, for which it

corresponds an eigenvector on the left r = (r1,..., ri,..., rn), such that ∀i, ri > 0.

Then, the converted system

( ) ( )[ ]iiiiiji Qrrqr , , (4)

6

Is such that its technological matrix A′

1ˆˆ' −= rArA = (a′ij)

has the property that

=∀i

ij aaj ' ,

And we demonstrate that we have all the properties a), b), c), d) and e) of the example of

Sraffa.

What we want to do is to follow the inverse way, to define the economical meaning of the

indecomposability with any size of the measurement units and, then, by modifying their

size through consecutive approximations, to convert the original system into another

structurally equivalent one, whose technological matrix A′ is such that

=∀i

ij aaj ' ,

maximum eigenvalue of A′ (and A), that admits )1,...,1(1 = as associated positive

eigenvector on the left, and from where the Perron-Frobenius theorem is deduced.

DECOMPOSABILITY AND GRAPH THEORY

We will say that the system (1) is decomposable –or reducible– if it exists a subset

{i1,…, ip}, 1 ≤ p < n of the system´s industries, such that these industries don´t use

products of the other industries. By negation we will say that the system is

decomposable –or irreducible– if any non-empty subset of industries –any industry

in particular i– uses directly or indirectly products of the other industries.

7

A system is mathematically decomposable if its technological matrix A –or qij– can be

converted through row and column changes to the form

=22

1211

AAA

A

in which A11 and A22 are square sub-matrices, being θ a null sub-matrix.

In the analysis that we propose it is very useful to interpret decomposability through

elemental graph theory.

DEFINITION. If N = {1,…, j,…, n} is the set of industries in the system, we will

denominate as graph the whole application Γ of N in the set P (N) of parts of N.

The elements of N are the vertex of the graph.

In a Leontief system with its technological matrix A = (aij), we define the graph as:

Γ (i) = { j ∈ N ⏐ aij ≠ 0 } ⊂ N

We will call arc of the graph Γ to every pair (i, j) of vertex (industries), such that j ∈

Γ(i); which is the same as saying that the industry j directly uses the product i.

We will say that i is the initial vertex and j the final one, simply denoting as i → j.

We can represent the N elements as points in the plane, in which case we describe Γ

through the set of its arcs.

For example, given the matrix

=1,06,02,04,001,02,003,0

A 33 2,3 ,13 },3 2, ,1{)3(

32 ,12 },3 ,1{)2(31 ,11 },3 ,1{)1(

→→→=Γ

→→=Γ

→→=Γ

We will call way to every series i1, i2,…, ik of vertex, such that ij+1 ∈ Γ(ij), i.e.,

i1 → i2 →… → ik–1 → ik

where the vertex i1 is the beginning of the way, and vertex ik is the end.

8

We will say that the industry j is accessible from i if it exists a way:

i = i1 → i2 →…→ ik–1 → ik = j

If aij ≠ 0, then i → j, which means that industry j directly uses the product i, while if

aij = 0 and j is accessible from i, the industry j = ik directly uses the product ik–1…

that uses i2, that uses the i1 = i; i.e. industry j indirectly uses product i.

We will say that the graph Γ of the system is strongly connected if for all industries i

and j there is a beginning of the way i and an ending of the way j, meaning that

industry j uses product i, directly if aij ≠ 0, or indirectly if aij = 0.

We will denominate circuit Ci to every way whose beginning i is confused with its

end in i: i = i1 → i2 →… → ik = i, and we will say that Ci is complete if

{i1, i2,…, ik} = {1,..., j…, n} = N

i.e. that starting from a vertex i it is possible to get “back” to i by a way that goes

through all vertex of the graph.

From previous conditions, we can give the following interpretation of the

decomposability of the system.

If it exists a complete circuit c (i) for all i , it means that each industry uses the

products of the other industries directly or indirectly, and that the system is

indecomposable.

EXAMPLE. For the matrix

=

0100002,001005,0008,00

A

3

2

4

1

9

the correspondent complete circuits are:

1 → 2 → 4 → 3 → 2 → 1

2 → 4 → 3 → 2 → 1 → 2

3 → 2 → 1 → 2 → 4 → 3

4 → 3 → 2 → 1 → 2 → 4

For each vertex i there is a complete circuit, so that A is indecomposable.

In general, each column j of A shows heterogeneous quantities of the goods directly

used in the production of an unit of good j, but the goods indirectly used and their

numerical quantities don´t appear.

In the previous example, industry 1 only uses the product of 2 which uses the products

of 1 and 3, using this last product the one of 4; that is to say, that industry 1 uses

products of the other industries direct or indirectly in its production, and we can say the

same about the industries 2, 3 and 4.

Let´s consider again the Leontief system that we described in (1) and a r = (r1,…, ri,…,

rn) ∈ Rn, such that ∀i, ri > 0, and modify the size of each unit i according to ri.

The system is converted into:

( ) ==

nn

ii

nn

ii

nnnnjnnn

iniijiii

nj

ij

Qr

Qr

Qr

Q

r

r

r

qrqrqr

qrqrqr

qrqrqr

q

= '

111 1

1

1

1 11 111 1

;

' ; ' (5)

The system (5) is structurally equivalent to (1), as each q′ij = ri qij represents the same

physical quantity of output i numerically expressed in different size units.

The technological matrix of the new system, ( ) '' ijaA = is:

ijj

i

jj

ijiij a

rr

Qrqr

a ==

'

10

And denoting as r̂ the diagonal matrix whose diagonal elements are the ri, we have 1ˆ ˆ ' −= rArA , deducing that the characteristical equations of A and A′

[ ]) ( detˆdet . ) ( det . ˆ det

ˆ )( ˆ det)ˆ ˆ(det ) ( det1

11

AIrAIrrAIrrArIA'I

−=−=

=−=−=−−

−− λλ

are the same, and thus that A and A′ have the same eigenvalues.

BANACH ALGEBRA Mn (R) AND PERRON-FROBENIUS THEOREM

The vectorial space Mn (R) of the square matrix of order n with the rule

=i

jij

aA sup

or, alternatively,

=j

jii

x aA sup

has Banach algebraic structure with unit I, being the rule compatible with the matrix

product2:

||A.B|| ≤ ||A|| . ||B||

||I|| = 1

Being A the technological matrix of a Leontief system and for each j, 1 ≤ j ≤ n, let´s say

=i

jij aS

2 See mathematical appendix.

11

We have, then

}{ sup jj

SA =

If all the Sj are equal to aA = , then a is the maximum eigenvalue of A that admits

)1,...,1,...,1(1 = as eigenvector on the left, and ∀λ ∈ R, λ > ||A||, we have that (λI – A) is

inversible and that (λI – A)–1 ≥ 03.

Given a technological matrix A with rule a = ||A||, and considering the matrix aA , with

rule 1. We will follow the reasoning with a matrix of rule 1

It is well known that if ||A|| < 1, I – A is inversible and

(I – A)–1 = I + A + A2 +…+ An +…

The problem of the inversibility of I – A appears when ||A|| = 1.

Now, we propose a version of the Perron-Frobenius theorem with a demonstration

based on topological considerations and the economic interpretation of the unit´s size

modification through consecutive approximations.

THEOREM. Being A a technological matrix such that

== 1 sup ijj

aA

if A is indecomposable, and at least for one j,

<=i

ijj aS 1

then I – A is inversible, its inverse is positive, and there exists r ∈ Rn, r > 0, such that

the matrix 1ˆ ˆ ' −= rArA verify that ∀j,

==i

ijj aaS ' '

3 See Mathematical appendix.

12

is a maximum eigenvalue of A, strictly minor than 1, and that admits )1,...,1(1 = as

associated eigenvector on the left.

DEMONSTRATION. Let´s demonstrate, first, that it exists r ∈ Rn, r > 0, such that

1ˆ ˆ 1 <−rAr .

Given ri ∈ R, ri > 0, we will denote ir the vector Rn whose components are all

equal to 1, except the one of order i , which is equal to ri.

Being i an industry such that

<=k

kii aS 1

If for every j ≠ i aij ≠ 0, it means that all industries j directly use the product i, so it

is enough to take ri, Si < ri < 1 and then, the addition of the elements of the columns j

≠ i of 1ˆ ˆ −rAr will be:

S′j = a1j +...+ aij ri +...+ anj < a1j +...+ aij + anj = Sj ≤ 1

And, as the same,

1...

...' 1 <++++=<i

ni

i

iii

i

iii r

ar

rara

SS

at the end, we have,

1}'{ supˆ ˆ 1 <=−j

jSrAr

Given that all the industries directly use the product i, when increasing the size of the

unit i a decrease in all Sj, j ≠ I, is caused.

13

If aij = 0, it means that the industry j does not directly use the product i but it is

indirectly used, as the system is indecomposable. Let´s consider, then, the complete

circuit:

i = i (1) → i (2) →… → i (t) → i (t+1) →…→ i (k) = i

in which i (t+1) is the first vertex for which Si (t+1) = 1.

Now, let´s take ri(t), such that Si(t) < ri(t) < 1 and that )(tir is the vector whose

components, except the one from order i (t), are equal to ri(t). The industry i (t+1)

directly uses the product i (t), so that the addition of the column elements i (t+1) of

1)()(

ˆ ˆ' −⋅⋅= titi rArA

will be

S′i(t+1) = a1,i(t+1) +…+ ai(t),i(t+1) ri(t) +…+ an,i(t+1) < Si(t+1) = 1

On the other side, given the election of ri (t), we also have S′k <1, k ≤ i (t). We can

restart the process from i (t+1), obtaining real numbers ri(t) , ri(t+1) ,…, ri(t+p).

Being r the vector of Rn whose components , except the one from order i (t), are

equal to ri(t) , those of order i (t+1) will be equal to ri(t+1)…

Through elemental calculations we have that

)()1()(ˆ ... ˆ ˆˆ ptititi rrrr ++ ⋅⋅⋅=

and 1ˆ ˆ ' −= rArA is such that

1}'{ sup' <= jj

SA

So that I – A′ is inversible, being its inverse positive, from where we deduce that the

eigevalues of A′ (and A) are strictly less than 1.

14

On the other side,

( ) ( ) ( )[ ] 1111111 ˆ )( ˆˆ ˆˆ ˆ' −−−−−−− −=−=−=− rAIrrAIrrArIAI

For simplicity, let´s denote A instead of A′

( ) ( )ijAI 1 =− − y =i

ijjr

We have, then,

( ) }{ sup 1j

jrAI =− −

If we consider the equations system r (I – A) = 1 , or, what it is equivalent, 1)( 1 −−= AIr , its solution is no other than r = (r1,..., rj,..., rn), where ij

ijr =

and, as (I – A)–1 = I + A +…+ An +… the different rj are strictly greater than 1.

From r (I – A) = 1 , we have that ∀j, 1 ≤ j ≤ n

– a1j r1 – a2j r2 –…+ (1 – ajj) rj –…– anj rn = 1

from where we obtain

j

nnj

j

jjj

jj

j rra

rr

arra

r+++++= ......11 1

1 (6)

Saying

1ˆ ˆ)1( −= rArA y )1( )1( =

iijj aS

15

we have that ∀j, 1 ≤ j ≤ n

)1( 11 jj

Sr

+=

If all rj are equal, that is to say, if the sum of the column elements of (I – A)–1 is equal

to r, we will have that the sum of the columns of A is equal to a , and that ar

+= 11 ,

from where a

r−

=1

1 , being, then, a the maximum eigenvalue of A.

From the main approach of the Leontief system, Q – A Q = β, and being I – A

inversible, Q = (I – A)–1 β:

Qi = αi1 β1 +...+ αij βj +...+ αin βn

If we increase the industry´s surplus in one unit j, it must happen an increase Δ Qi in

the production of the industry i such that

Qi + Δj Qi = αi1 β1 +...+ αij (βj+1) +...+ αin βn

so that Δj Qi = αij represents the direct and indirect increase in the production of i

following an increase of an unit of j, and being the system indecomposable, Δj Qi = αij

≠ 0.

The sums

ij

iiijj Qr Δ==

represent heterogeneous quantities of the goods that the system must produce when βj

increases in one unit. There it is the importance of considering the size of this unit

relative to the size of the other units. So that a small ri means that the unit is small

relative to the different sizes.

16

As in the columns of (I – A)–1 there are heterogeneous quantities of the goods directly

and indirectly used by each industry, we suggest the possibility of using the sums

=i

ijjr

as the elements of the original system transformed into another, structurally equivalent,

one.

Let´s denote 1)()(~ −−== AIA ij , and suppose it exists at least one j such that

Ai

ij~ <

From 11 ˆ )( ˆˆ ˆ −− −=− rAIrrArI we also have

( ) 11111 ˆ ~ ˆˆ )( ˆˆ ˆ −−−−− =−=− rArrAIrrArI

Given that by construction

=i

ijjr

the following matrix

( ) 11 ˆ )( −−−= rAIij

is such that ∀j, 1 ≤ j ≤ n

=i

ij 1

And the sum of each column j of

( ) 1ˆ ~ ˆ ˆ −= rArr ij

17

is

Arrrr iii

ijiii

ijii

ijii

~}{ sup }{ sup }{ inf ==<<

from where

ArrArr ii

ii

~}{ sup ˆ ~ ˆ }{ inf 1 <<< −

being the rule of the transformed matrix strictly minor than the rule of the original one.

Let´s say, then, that

1

1

~)}1( { sup (1)

)}1( { inf (1) )1( ~ 1)1( ˆ

ˆ ~ ˆ)1( ~

Ar

rAr

rArA

ii

ii

==

==

= −

Through an identical reasoning, we have

(1) )1( ˆ )1( ~ )1( ˆ)1( 1 << −rAr

Given that 1ˆ ~ ˆ 1)1( ˆ −= rArr , we have that ∀j, 1 ≤ j ≤ n

( ) }{ inf...

}{ inf ... 1)1( 111 ii

j

njjiinnjj

jj r

rrrr

rr =

++>++=

so that,

=> )0( )1( }{ inf iir and A~)0( )1( =<

Repeating the reasoning we obtain a matrix series

)( ~ ,..., )1( ~ , ~)0( ~ nAAAA = ,...

18

and two series of real numbers: ( ))( n strictly increasing and upper-bounded and thus

convergent: )( lim n= ; and ( ))( n strictly decreasing and lower-bounded, and thus

also convergent: )( lim n= .

Now we will demonstrate that being A~ indecomposable, necessarily = . By

construction, the different )( ~ nA are indecomposable, and the change of )( ~ nA into

1)( ˆ )( ~ )( ˆ)1( ~ −=+ nrnAnrnA is done by jointly readjusting the size of the different

units, as each product directly and indirectly goes into the production of the other

products.

We will demonstrate that necessarily

)( lim)( lim nn ===

Firstly, let´s observe that with a given vector r ∈ Rn of strictly positive components,

}{ inf11 supˆ

}{ supˆ

1

iiii

ii

rrr

rr

==

=

−

from where

rrr

ˆ1ˆˆ 11 == −− if and only if

}{ sup1

}{ inf1

ii

iirr

=

That is to say, only if all the ri are equal.

Then,

[ ]1

1

)( ˆ . )( ~)( ˆ lim

)( ˆ)( ~)( ˆ lim)1( ~ lim)1( lim−

−

⋅≤

≤⋅⋅=+=+=

nrnAnr

nrnAnrnAn

19

because Mn (R) is a Banach algebra, and the rule of a matrix product is minor or equal

to the rules´product, and given that,

)( lim)( ~ lim == nnA

we must have

11 )( ˆ lim)( ˆ lim −− = nrnr

That is to say,

1 = 1)( −

or, because it is the same, = and then,

)'()( ~ lim'~ijnAA ==

is such that ∀j, 1 ≤ j ≤ n

=i

ij'

We have then, maximum eigenvalue of

)( ~ lim'~ nAA =

and given that, by construction, setting

1)( ˆ )( )( ˆ)1( −=+ nrnAnrnA

20

we also have

( ) )( ~)( 1 nAnAI =− −

Inverting both sides, it results

1)( ~)( −−= nAInA

so that

( ) 11 '~)( ~ lim)( lim' −− −=−== AInAInAA

and the maximum eigenvalue a of A′ is

11−=a

Furthermore, and also by construction, matrices ( )ijA ''~ = and A′ = (aij) are such that

∀j, 1 ≤ j ≤ n

=i

ij' y =i

ij aa

That´s because they admit as positive eigenvector on the left )1,...,1(1 = or any positive

multiple of it.

The economic interpretation of the procedure is easier to see now. Each column j of

'~A represents heterogeneous quantities of the system´s goods needed to produce an unit

of j as final good, and given that all the columns sum , in order to produce an

additional quantity of any good, we need the same numerical quantity of the system´s

heterogeneous goods, so that an unit of good i must be equivalent to another of good j,

i.e. all the units can be expressed in terms of one of them –or a multiple or part of it–

chosen as numeraire.

21

With the target of illustrating the consecutive approximation procedure, let´s take again

the simple example of Sraffa relative to the wheat and iron industry, in which the wheat

unit is the quarter and the iron unit is the ton.

The technological matrix was

=4,0020869,0

6486956,0A

For which S1(0) ≈ 0,507825 and S2 (0) = 6,4, as the quarter is an excessively small-sized

unit in comparison with the ton (or the ton excessively larger than the quarter).

Leontief´s inverse:

≅−= −

809475,211428,0856548,32285654,3

)()0( ~ 1AIA

from where r1 (0) ≅ 3,4 and r2 (0) ≅ 35,666023.

If we say now

1

1 )0( ˆ)( )0( ˆ)1( ~ −−−= rAIrA

we obtain

≅809475,2198794,1

1321788,3285654,3)1( ~A

from where r1 (1) = 4,484448 and r2 (1) = 5,941653 and

≅= −

809475,2588323,1364003,2285654,3

)1( ~ )1( ˆ)2( ~ 1 )1( r̂ArA

where r1 (2) = 4,873977 and r2 (2) = 5,173478.

22

In the following stage we would obtain r1 (3) = 4,971577 and r2 (3) = 5,036622. We can

see that the series )( )( 1 nrn = and )( )( 2 nrn = tend to = = 5, and that the

maximum eigenvalue of matrix A′ converted into A would be 8,0511 =−=a , as we

expected.

We can also observe that in this case of two unique industries

1594,1445,36984,5521

)3( )2( )1( )0( )3( )2( )1( )0(

1111

2222 ≅=≅⋅⋅⋅⋅⋅⋅rrrrrrrr

That was the exchange relation in the original system:

1 iron ton = 15 wheat quarters

HOMOGENEIZATION OF THE UNITS

We have demonstrated that if the economic system has any kind of surplus and it is

irreductible, the máximum eigenvalue of A, a , is strictly minor than 1 and the system

can be transformed into another equivalent where its technological matrix A′= (a′ij) is

such that ∀j, 1 ≤ j ≤ n

=i

ij aa'

Being )1,...,1(1 = eigenvector on the left of A′

S ∈ Rn, S > 0 (∀i, Si > 0), and denoting S1 vector

iiS1

matrix 1ˆ ' ˆ)1( −= SASA has the same eigenvalues as A′ and, as SASA ˆ )1( ˆ ' 1−= and

SS 1ˆ 1 1 =− , of 1 ' 1 aA = we deduce that ( ) 11 1)1( ˆ 1 −− ⋅=⋅ SaAS , i.e.

SaA

S1)1( 1 =

and S1 is an eigenvector on the left of A (1).

23

A (1) (S ∈ Rn, S >0) are a family of matrices of Mn (R) representative of the Leontief

system set. In particular, given the original matrix A, it exists

n

Sr R1 ∈= , r > 0

such that raAr = , being r the eigenvector on the left of A.

Let´s take again system (4) and modify the size of the units in equal terms as ri

components of vector r.

From raAr = we have that ∀j, 1 ≤ j ≤ n

r1 aij +…+ ri aij +…+ rj ajj +…+ rn anj = jra (7)

so that

.........1 aarra

rr

arra

rr

njj

njj

j

jij

j

iij

j

=++++++

And setting A′ = (a′ij) with

ijj

iij a

rra ='

1ˆˆ' −= rArA

and ∀j, 1 ≤ j ≤ n

=i

ij aa'

DEFINITION. Given an indecomposable Leontief system, we will call homogenized

system the transformed equivalent system whose technological matrix A is such than

∀j,

=i

ij aa

and we will call A the homogenized matrix.

24

Among the infinite technological matrices A (S) representative of the same system, it

will be useful to take as representative the homogenized one, in particular when

referring to the economic interpretation of the main matrix A and its Leontief inverse.

System (7) is equivalent to 0) ( =− AIar (an homogeneous system of n equations

and n unknowns with range n–1 (as a is the simple root of the characteristical

equation). For its resolution we can take one of the original units (or part of them) as

numeraire, for example, rn = 1.

We will transform the system such that all the units will be expressed in terms of rn

= 1, so they will behave as single product units inside the system. Then, the

heterogeneous quantities that appear in each column j of A behave as if they were

homogeneous.

THE LEONTIEF INVERSE

Being A homogenized, we have that ∀j,

Aaai

ij ==

Let´s set, then, (γij) = A2, and we have

kjk

ikji aa =

2 aaaaaaak

kji i k

kik

kjk

kjikji =⋅===

That is to say, 22 AA = , from where we deduce that ∀n nn AA = and the sum

of the columns of An are all equal, and equal to na .

Then, as

(I – A)–1 = I + A +...+ An + ...

25

The sum of any column elements of (I – A)–1 will be

aaa n

−=++++

11......1

We have previously seen that each column j of (I – A)–1 represents heterogeneous

quantities of goods needed for the production of an additional unit of j and, as all

column sums are the same in the homogenized system, this means that we need the

same quantities of goods to produce an additional unit of any of them.

THE STRUCTURAL RATIO

If the system has previously been homogenized, matrix A = (aij) is such that ∀j, 1 ≤ j ≤ n

=i

ij aa

or even

ji

ij Qaq =

Adding both sides over j,

= jjji

ij Qaq ,

or, if its preferred,

aQ

q

jj

jiij

=

, (8)

26

That is to say, the ratio between the total quantity of goods used in production and the

total quantity of goods produced is equal to the maximum eigenvalue a .

From (8), and taking into account that

+=j

jji

ijj

j qQ ,

we also deduce

aQj

jj

−= 1

j

(8′)

11

−= aqiji,j

jj

(8′′)

We obtain, as function of a , ratios among surplus produced, and the total production of

the system. We will say that (8) is the system´s structural ratio.

MATHEMATICAL APPENDIX

We will denote as Mn (R) the vectorial space of the square matrix of order n with real

coefficients. In Mn (R) we also have another internal composition rule, the square

matrix product: if A = (aij) ∈ Mn (R) and B = (bij) ∈ Mn (R), A.B = (γij), with (γij) =

kkjik ba . The product is associative, it is distributive over matrix addition, and it admits

as neutral element the identity matrix I.

THEOREM 1. In Mn (R) if A = (aij), setting =i

ijj

aA sup , ⋅ is a rule in Mn (R),

and thus Mn (R) is a Banach algebra.

Proof:

a) That =i

ijj

aA sup is a rule in Mn (R) is obvious.

27

b) Mn (R) is complete. So, being (At)t = ( )t

tija a Cauchy series in Mn (R) ,

∀ε >0, ∃ t0 ∈N / p ≥ t0, q ≥ t0 ε<− qp AA

but

<−⇔<−=

n

i

qij

pij

j

qp aaAA1

sup εε

And in particular, being i and j fixed, we have ε<− qij

pij aa , so that the series of real

numbers ( )t

tija is Cauchy in R and converge to a real number that we will denote as

aij.

Setting then, A = (aij), it is enough to demonstrate that (At)t converges to A in Mn (R).

Be ε ∈ R, ε>0, as ( )t

tija converges in R to aij,

∃ t (i, j) ∈ N / t ≥ t (i, j) n

aa ijtij

ε<−

And if we take t0 = sup { }njnijit ≤≤≤≤ 1 , 1 ),,( , then

∀t ≥ t0 y ∀j , 1 ≤ j ≤ n, ε=ε⋅<−= n

naan

iij

tij

1

From where it results ε<−=

sup1

n

iij

tij

jaa or even ε<− AAt , so that (At)t

converges to A. Mn (R) is a Banach space.

c) ∀ A, B ∈ Mn (R), BABA ⋅≤⋅ and 1=I , if A = (aij) and B = (bij), setting

A.B = (γij), with (γij) = k

kjik ba we have:

BAAab

babaBA

iij

ji kkiij

j

kkjik

ijikjik

jiij

jij

b ⋅≤≤

≤≤===⋅

=

sup . sup

sup sup sup )( γγ

The chosen rule is “compatible” with the matrix product. So it is obvious that 1=I .

THEOREM 2. For A ∈ Mn (R) to be invertible in Mn (R), it is necessary and sufficient

that it exists D ∈ Mn (R), D inversible and such that 1 1 <=− − KDAI .

28

Proof: If A is inversible, we take D = A, we have 10 1 <=⋅− −AAI . In both sides, we

suppose that it exists D inversible, such that 1 1 <=− − KDAI , the application

Φ : Mn (R) → Mn (R) :

φ (x) = D–1 + x ( )1 −− DAI

is such that

( ) 11 ' )'( )( )( −− −⋅−≤−−=′− DAIxxDAIxxxx φφ

(because Mn (R) is a Banach algebra), and as 1 1 <=− − KDAI , Φ is a complete

contraction in Mn (R) , and admits an unique fixed point x .

We have ( ) xx = φ or

( ) xDAIxD =−+ −− 11 , x A D–1 = D–1,

and multiplying by D on the right, x A = I and x is the inverse of A.

Moreover, ∀ x0 ∈ Mn (R), the series x0, x1= φ (x0),..., xn = φ (xn-1),... converges at

x =A–1 (fixed point theorem).

Definition. Being A = (aij) ∈ Mn (R), we will say that A is column dominant diagonal

if ∀ j, 1 ≤ j ≤ n, >≠ ji jjj iaa ; we will also say that A is Leontief if

≠≤

>∀

jia

ai

ij

ii

si 0

0 ,

Obviously, if A is the technological matrix in a Leontief system, I – A is Leontief.

THEOREM 3. If A = (aij) ∈ Mn (R) if Leontief´s column dominant diagonal, A is

inversible in Mn (R) and also, A–1 ≥ 0.

Proof. Being A dominant diagonal, ∀j, 1≤j≤n,

>≠ ji

ijjj aa y <=≠ ji

ijjj

kaa

1 1

Let´s consider the matrix D defined by djj = ajj, dij = 0 if i ≠ j. As djj = ajj > 0, D is

inversible in Mn (R) and also,

1 1

sup 1 <≤=−≠

− kaa

DAIji

ijjjj

29

so that the application

Φ : Mn (R) → Mn (R) :

φ (x) = D–1 + x ( )1 −− DAI

is a contraction in Mn (R) and it admits an unique fixed point x = A–1. Moreover,

given that D–1 ≥ 0 and ( )1 −− DAI ≥ 0, starting from x0 ≥ 0, the row x0, x1=φ (x0),...,

xn=φ (xn–1),... is such that ∀n, xn ≥ 0 and, consequently, 0 lim 1 ≥== −

∞→Axxnn

.

THEOREM 4. If A ∈ Mn (R) and λ is eigenvalue of A, we have A≤λ .

Proof. If λ is eigenvalue of A, λ I – A is not inversible and then, (theorem 2), for the

whole inversible matrix D we have ( ) 1 1 ≥−− −DAII λ . If λ = 0, this means that A is

not inversible and, obviously, A≤= 0λ . If λ ≠ 0, and considering the diagonal

matrix D = (dij) defined by dii=λ, dij = 0 if i≠j, we have:

( ) AiaDAIIn

i jj

1

1

sup 1

λλλ ==−− −

And as it must be 1 1

≥Aλ

, it results A≤λ .

THEOREM 5. Being A ∈ Mn (R), and A ≥ 0

a) If ∀ j, 1 ≤ j ≤ n, ===

n

iij Aaa

1 , then a is maximum real eigenvalue of A that

admits 1 = (1,...,1) as associated eigenvector on the left.

b) If λ ∈ R is such that λ > A , then λ I – A is inversible and (λ I – A)–1 ≥ 0.

Proof.

a) It results that 1 A = a 1 , and a is a real eigenvalue. On the other side, as for every

eigenvalue λ, A≤λ (theorem 4), it results that a is the maximum eigenvalue.

30

b) Let´s consider matrix D = (dij) defined by dii = λ and dij = 0, if i ≠ j. We have

that ( ) 1 1 <=−− −

λλ

ADAII and the application

Φ : Mn (R) → Mn (R) :

φ (x) = D–1 + x ( )( )1 −−− DAII λ

is a contraction in Mn (R) that admits an unique fixed point ( ) 1 −−= DAIx λ (theorem

2) and, given that D–1 ≥ 0 and ( ) 1 −−− DAII λ ≥ 0, starting from x0 ≥ 0, the row x0, x1

= φ (x0),..., xn = φ (xn–1),... is such that ∀n xn ≥ 0 and x = (λ I – A)–1 = lim xn ≥ 0.

BIBLIOGRAPHY

Debreu, G.; Herstein, I.N. (1953): “Non Negative Square Matrices”, Econometrica, 21, pp. 597-607.

Gantmacher (1966): Thèorie de matrices, t. 2. Paris: Dunod.

Leontief, V. (1951): The Structure of American Economy 1919-1929. New York: Oxford University Press.

Maddox, J.J. (1980): “Infinite Matrices of Operators”, Lectures Notes in Mathematics, 786. Springer Verlag.

McKenzie, L. (1959): “Matrices with Dominant Diagonals and Economic Theory”, en Arrow, Karlin y Suppes [ed.]: Mathematical Methods in the Social Sciences. Stanford University Press.

Pasinetti, L. (1983): Lecciones de teoría de la producción. Fondo de Cultura Económica.

Quiñoá, J.L. (1983): Inversibilidad en álgebras de Banach de matrices infinitas y aplicación a los sistemas de ecuaciones de orden infinito. (Tesis doctoral). Zaragoza.

Quiñoá, J.L. (1992): “Sur un type de matrice infinie de diagonale dominante dans la thèorie économique”, European Meeting of the Econometric Society. Bruselas.

QUIÑOÁ, J.L. (2011): Topologías en espacios de matrices y sistemas Leontief y Leontief-Sraffa. (http://EconPapers.repec.org/paper).

QUIÑOÁ, J.L.; Pie, L. (2011): “Homogenization in a Leontief (or Leontief-Sraffa) Type System”, GSTF Business Review, 1 (1).

Sraffa, P. (1960): Production of comodities by means of comodities . Cambridge University Press.

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